`log_3x=log_9(6x)` Solve the equation.

Expert Answers

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To solve the equation `log_3(x)=log_9(6x)`, we may apply logarithm properties.

Apply the logarithm property: `log_a(b)= (log_c(b))/log_c(a)` on `log_3(x)` , we get:


Let `3 =sqrt(9) = 9^(1/2)`


Apply the logarithm property: `log(x^n)= n*log(x) ` and `log_a(a)=1 ` on `log_9(9^(1/2))` .






Apply the logarithm property: `log(x*y)=log(x)+log(y)` on `log_9(6x)` .





Apply the logarithm property:`a^(log_a(x))=x` on both sides.



Check: Plug-in `x=6` on `log_3(x)=log_9(6x).`




Final Answer:

`x=6` is a real solution for the equation `log_3(x)=log_9(6x)` .

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